let f1, f2 be PartFunc of REAL , REAL ; :: thesis: ( f1 is divergent_in-infty_to+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st
( g < r & g in dom (f2 * f1) ) ) implies ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_in+infty f2 ) )
assume A1: ( f1 is divergent_in-infty_to+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st
( g < r & g in dom (f2 * f1) ) ) ) ; :: thesis: ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_in+infty f2 )

A2: now

let s be Real_Sequence; :: thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) )
assume A3: ( s is divergent_to-infty & rng s c= dom (f2 * f1) ) ; :: thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 )
then A4: ( rng s c= dom f1 & rng (f1 /* s) c= dom f2 ) by Lm2;
then A5: f1 /* s is divergent_to+infty by A1, A3, LIMFUNC1:def 10;
lim_in+infty f2 = lim_in+infty f2 ;
then ( f2 /* (f1 /* s) is convergent & lim (f2 /* (f1 /* s)) = lim_in+infty f2 ) by A1, A4, A5, LIMFUNC1:def 12;
hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) by A3, VALUED_0:31; :: thesis: verum

end;

hence f2 * f1 is convergent_in-infty by A1, LIMFUNC1:def 9; :: thesis: lim_in-infty (f2 * f1) = lim_in+infty f2
hence lim_in-infty (f2 * f1) = lim_in+infty f2 by A2, LIMFUNC1:def 13; :: thesis: verum